If p ≠ 0 and q ≠ 0 are the roots of the equation x² + px + q = 0, then what is p² + q² equal to?
5
Problem
If p ≠ 0 and q ≠ 0 are the roots of the equation x² + p x + q = 0, find p² + q².
Solution (step-by-step)
- By Vieta's formulas for x² + p x + q = 0, the sum of the roots equals -p and the product equals q.
- But the roots themselves are p and q. Hence
p + q = -p and p q = q
- From p q = q we get q (p - 1) = 0. Since q ≠ 0, it follows that p = 1.
- Substitute p = 1 into p + q = -p:
1 + q = -1 so q = -2
Finally compute
p² + q² = 1² + (-2)² = 1 + 4 = 5
Final answer: p² + q² = 5
Similar Questions
(x + 4) is a factor of which one of the following expressions?
If \(\sqrt {\frac{{\rm{x}}}{{\rm{y}}}} = \frac{{24}}{5} + \sqrt {\frac{{\rm{y}}}{{\rm{x}}}} \) and x + y = 26, then what is the value of xy?
If α and β are the roots of the equation x 2+ px + q = 0, then what is α 2+ β 2equal to?
It is given that the equations x 2– y 2= 0 and (x – a) 2+ y 2= 1 have single positive solution. For this, the value of ‘a’ is
If x = 2 + 2 2/3 + 2 1/3 , then what is the value of x 3– 6x 2+ 6x?
If p and q are the roots of x 2+ px + q = 0, then which of the following is correct?
If 4x + 3a = 0, then what is the value of \(\frac{{{x^2}\; + \;ax\; + \;{a^2}}}{{{x^3} - {a^3}}} - \;\frac{{{x^2} - \;ax\; + \;{a^2}}}{{{x^3}\; + \;{a^3}}}\;\) ?
Let p and q be non-zero integers. Consider the polynomial A(x) = x 2+ px + q. It is given that (x − m) and (x − km) are simple factors of A(x), where m is a non-zero integer and k is positive integer, k ≥ 2. Which one of the following is correct?
If α and β are the roots of the equation x + a + b = abx, then what is (αβ + α + β) equal to?
If p = 0.09, then what is the value of 70p² + 43p - 5?
Important Questions from Quadratic Equations
If the graph of a quadratic polynomial lies entirely above x-axis, then which one of the following is correct?
If the roots of the equation x 2+ px + q = 0 are tan 19° and tan 26°, then which one of the following is correct?
If x 2- px + 4 > 0 for all real values of x, then which one of the following is correct?
If 2 + i is a root of the equation x 2- ax + 1 = 0, then the value of a is:
If \(\tan \left( {\frac{α }{2}} \right)\) and \(\tan \left( {\frac{β }{2}} \right)\) are the roots of the equation 8x 2- 26x + 15 = 0, then the value of cos (α + β) will be
